A theory of fractal fluids

Practical fluid mechanic problems may be characterized by a length scale epsilon, below which we are not able or not willing to resolve details of density or velocity distributions. Conventional fluid mechanics assumes that below the epsilon-scale all essential quantities (in particular the density and velocity) can be considered as continuous and are accurately enough described by their average values taken over control volumes of size epsilon. This assumption, however, is not valid for many problems of interest. The category of these problems includes both turbulent flow and two-phase flow. The fundamental property of these fluids is that local flow characteristics (such as density or velocity) at any given point must be considered to be different from any (possibly weighted) average of this quantity in a small control volume surrounding this point, even for the smallest control volumes we can practically choose. This paper presents a new method to deal with such problems of fluid mechanics. The method is based on a rigorous treatment of the transport theorem and uses Taylor series expansions (with third order accuracy in the spatial variables) of the essential functions whenever exact solutions could not be obtained. A closed system of partial differential equations is derived. It contains conventional fluid dynamics as a special case as well as results which can be derived from the kinetic theory of gases. (orig./GL)